Unit Circle

Evaluate the integral of cos(2x) divided by the square root of (1-sin^2(2x)) with respect to x

To evaluate the integral $ \int \frac{\cos(2x)}{\sqrt{1-\sin^2(2x)}} \, dx $, we begin by recognizing that:

$$ \sin^2(2x) + \cos^2(2x) = 1 $$

Thus, the expression under the square root simplifies to:

$$ \sqrt{1-\sin^2(2x)} = \cos(2x) $$

Substituting this into the integral gives:

$$ \int \frac{\cos(2x)}{\cos(2x)} \, dx $$

This simplifies to:

$$ \int 1 \, dx $$

The integral of 1 with respect to $x$ is:

$$ x + C $$

Find the secant of an angle θ in a unit circle

To find the secant of an angle $\theta$ in a unit circle, we use the formula:

$$ \sec(\theta) = \frac{1}{\cos(\theta)} $$

Suppose $\theta$ is an angle in the first quadrant where cos(θ) = 0.6. Then:

$$ \sec(\theta) = \frac{1}{0.6} = \frac{5}{3} $$

Calculate the tangent of an angle when given the sine and cosine values in the unit circle

To find the tangent of an angle in the unit circle when given the sine and cosine values, we use the formula:

$$ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} $$

For example, if $\sin(\theta) = \frac{1}{2}$ and $\cos(\theta) = \frac{\sqrt{3}}{2}$, then:

$$ \tan(\theta) = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3} $$

Determine the values of cos(theta) and sin(theta) given that the point (x, y) is on the unit circle

Given that $ (x, y) $ is on the unit circle, we know:

$$ x^2 + y^2 = 1 $$

Using the definitions of the trigonometric functions on the unit circle, we have:

$$ \cos(\theta) = x $$

$$ \sin(\theta) = y $$

Thus, the values of $ \cos(\theta) $ and $ \sin(\theta) $ are:

$$ \cos(\theta) = x $$

$$ \sin(\theta) = y $$

Define a unit circle and prove that any point (x, y) on the unit circle satisfies the equation x^2 + y^2 = 1

A unit circle is a circle with a radius of 1 unit, centered at the origin (0,0) in the Cartesian coordinate system.

To prove that any point $ (x, y) $ on the unit circle satisfies $ x^2 + y^2 = 1 $, we start with the definition of a circle:

$$ (x – h)^2 + (y – k)^2 = r^2 $$

For a unit circle, the center is at (0, 0) and the radius $ r $ is 1, so the equation becomes:

$$ x^2 + y^2 = 1 $$

Thus, any point $ (x, y) $ on the unit circle will satisfy this equation.

Find the exact value of sin(π/4) on the unit circle

To find the exact value of $ \sin(\frac{\pi}{4}) $ on the unit circle, we recognize that $ \frac{\pi}{4} $ is equivalent to $ 45^{\circ} $.

On the unit circle, the coordinates for $ \frac{\pi}{4} $ are $ \left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right) $.

The sine value is the y-coordinate, so:

$$ \sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} $$

Find the value of sin(θ) and cos(θ) where θ = π/3 using the unit circle

To find the values of $ \sin(\theta) $ and $ \cos(\theta) $ where $ \theta = \frac{\pi}{3} $, we use the unit circle.

On the unit circle, the coordinates of the point corresponding to $ \theta = \frac{\pi}{3} $ are:

$$ \left(\cos\left(\frac{\pi}{3}\right), \sin\left(\frac{\pi}{3}\right)\right) $$

From the unit circle, these values are:

$$ \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} $$

$$ \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} $$

Find the value of tan(x) for x = pi/4

To find the value of $ \tan(x) $ when $ x = \frac{\pi}{4} $, we use the unit circle chart.

For $ x = \frac{\pi}{4} $, the coordinates on the unit circle are (\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}).

The tangent function is defined as:

$$ \tan(x) = \frac{\sin(x)}{\cos(x)} $$

So,

$$ \tan\left( \frac{\pi}{4} \right) = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1 $$

Explain the concept of a unit circle, including its importance in trigonometry and how it relates to the coordinates of points on the circle

A unit circle is a circle with a radius of 1, centered at the origin of a coordinate system. The equation of the unit circle is given by:

$$ x^2 + y^2 = 1 $$

The unit circle is fundamental in trigonometry as it defines the sine and cosine functions for all real numbers. For any angle $\theta$, the coordinates of the corresponding point on the unit circle are $(\cos(\theta), \sin(\theta))$. These coordinates are derived from the definitions:

$$ \cos(\theta) = \frac{x}{1} = x $$

$$ \sin(\theta) = \frac{y}{1} = y $$

Additionally, the unit circle helps in visualizing and understanding periodic properties of trigonometric functions and their symmetries.

Find the angle in degrees corresponding to 7π/6 radians on the unit circle

To convert $\frac{7\pi}{6}$ radians to degrees, we use the conversion factor:

$$ 180^{\circ} = \pi \text{ radians} $$

Thus,

$$ \frac{7\pi}{6} \times \frac{180^{\circ}}{\pi} = 210^{\circ} $$

The angle in degrees is:

$$ 210^{\circ} $$